This paper establishes tractable bounds of joint source-channel coding with hierarchical sources in the finite blocklength regime. In this setting, both the indirect source and observable source must be reconstructed under correlated distortion constraints, leading to a joint excess-distortion event. First, to build computable tight bounds, we introduce a novel $\mathsf{d}(\cdot)$-functional distortion relaxation, which enables tractable and tight bounding of the joint excess-distortion probability induced by correlated sources. By this approach, the nonasymptotic converse and achievability bounds are given. Second, Gaussian approximations for the proposed bounds are obtained, which are optimal for the transmission of a Gaussian memoryless source over an additive white Gaussian noise channel with mean-square error distortion. The optimal scheme is obtained via a structured analysis that captures the intrinsic tradeoff between semantic and observable reconstructions. Furthermore, for the transmission of Gaussian memoryless sources over AWGN channels, we obtain explicit and computable bounds, by providing a new geometric structure involving three correlated spherical regions. This results extend the classical two-spherical region analysis for a single distortion constraint. Numerical simulations demonstrate that the proposed achievability and converse bounds tightly sandwich the Gaussian approximation and align closely with Monte Carlo numerical results.

翻译：本文建立了有限码长条件下分层信源的联合信源信道编码的可处理界。在此设定中，间接信源和可观测信源需在相关失真约束下进行重构，从而导致联合超失真事件。首先，为构建可计算的紧致界，我们引入了一种新颖的$\mathsf{d}(\cdot)$-泛函失真松弛方法，该方法能够对相关信源引起的联合超失真概率进行可处理且紧致的界定。基于此方法，给出了非渐近的逆界和可达界。其次，获得了所提出界的高斯近似，该近似在高斯无记忆信源通过加性白高斯噪声信道传输且采用均方误差失真度量的场景下达到最优。通过一种捕捉语义重构与可观测重构之间内在权衡的结构化分析，得到了最优方案。此外，针对高斯无记忆信源通过AWGN信道的传输问题，我们利用涉及三个相关球面区域的几何新结构，获得了显式且可计算的界。这一结果将针对单一失真约束的经典两球面区域分析进行了推广。数值仿真表明，所提出的可达界与逆界能够紧密地夹逼高斯近似，并与蒙特卡洛数值结果高度吻合。